Woodin cardinals (named after W. Hugh Woodin) are a generalization of the notion of strong cardinals and have been used to calibrate the exact proof-theoretic strength of the axiom of determinacy. They can also be seen as weakenings of Shelah cardinals, defined below. Their exact definition has several equivalent but different characterizations, each of which is somewhat technical in nature. Nevertheless, an inner model theory encapsulating infinitely many Woodin cardinals and slightly beyond has been developed.

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Definition and some properties

We first introduce the concept of $\gamma$-strongness for $A$: an ordinal $\kappa$ is $\gamma$-strong for $A$ (or $\gamma$-$A$-strong) if there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{\kappa+\gamma}\subseteq M$ and $A\cap V_{\kappa+\gamma} = j(A)\cap V_{\kappa+\gamma}$. Intuitively, $j$ preserves $A$.

We also introduce Woodin-ness in $\delta$: for an infinite ordinal $\delta$, a set $X\subseteq\delta$ is Woodin in $\delta$ if for every function $f:\delta\to\delta$, there is an ordinal $\kappa\in X$ with $\{f(\beta)|\beta<\kappa\}\subseteq\kappa$, there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)}\subseteq M$.

An inaccessible cardinal $\delta$ is Woodin if any of the following (equivalent) characterizations holds [1]:

For any set $A\subseteq V_\delta$, there exists a $\kappa<\delta$ that is $\gamma$-strong for $A$ for every $\gamma<\kappa$.

For any set $A\subseteq V_\delta$, the set $S=\{\kappa<\delta|\kappa$ is $\gamma$-strong for $A$ for every $\gamma<\kappa\}$ is stationary in $\delta$.

The set $F=\{X\subseteq \delta|\delta\setminus X$ is not Woodin in $\delta$$\}$ is a proper filter, the Woodin filter over $\delta$.

For every function $f:\delta\to\delta$ there exists $\kappa<\delta$ such that $\{f(\beta)|\beta\in\kappa\}\subseteq\kappa$ and there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\kappa$ such that $V_{j(f)(\delta)}\subseteq M$.

Let $\delta$ be Woodin, $F$ be the Woodin filter over $\delta$, and $S=\{\kappa<\delta|\kappa$ is $\gamma$-strong for $A$ for every $\gamma<\kappa\}$. Then $F$ is normal and $S\in F$. [1] This implies every Woodin cardinal is Mahlo and preceeded by a stationary set of measurable cardinals. However, Woodin cardinals are not weakly compact as they are not $\Pi^1_1$-indescribable.

Woodin cardinals are weaker consistency-wise then superstrong cardinals. In fact, every superstrong is preceeded by a stationary set of Woodin cardinals.

The existence of a Woodin cardinal implies the consistency of ZFC + "the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturared". Huge cardinals were first invented to prove the consistency of the existence of a $\aleph_2$-saturated ideal on $\omega_1$, but turned out to be stronger than required, as a Woodin is enough.

Shelah cardinals

Shelah cardinals were introduced by Shelah and Woodin as a weakening of the necessary hypothesis required to show several regularity properties of sets of reals hold in the model $L(\mathbb{R})$ (e.g., every set of reals is Lebesgue measurable and has the property of Baire, etc...). In slightly more detail, Woodin had established that the axiom of determinacy (a hypothesis known to imply regularity properties for sets of reals) holds in $L(\mathbb{R})$ assuming the existence of a nontrivial elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point $<\lambda$. This axiom, a rank-into-rank axiom, is known to be very strong and its use was first weakened to that of the existence of a supercompact cardinal. Following the work of Foreman, Magidor and Shelah on saturated ideals on $\omega_1$, Woodin and Shelah subsequently isolated the two large cardinal hypotheses which bear their name and turn out to be sufficient to establish the regularity properties of sets of reals mentioned above.

Shelah cardinals were the first cardinals to be devised by Woodin and Shelah. A cardinal $\delta$ is Shelah if for every function $f:\delta\to\delta$ there exists a nontrivial elementary embedding $j:V\to M$ with critical point $\delta$ such that $V_{j(f)(\delta)}\subseteq M$. Every Shelah is Woodin, but not every Woodin is Shelah: indeed, Shelah cardinals are always measurable and in fact strong, while Woodins are usually not. However, just like Woodins, Shelah cardinals are weaker consistency-wise than superstrong cardinals.

A related notion is Shelah-for-supercompactness, where the closure condition $V_{j(f)(\delta)}\subseteq M$ is replaced by $M^{j(f)(\delta)}\subseteq M$, a much stronger condition. The difference between Shelah and Shelah-for-supercompactness cardinals is essentially the same as the difference between strong and supercompact cardinals, or between superstrong and huge cardinals. Also, just like every Shelah is preceeded by a stationary set of strong cardinals, every Shelah-for-supercompactness cardinal is preceeded by a stationary set of supercompact cardinals. Woodin-for-supercompactness cardinals were also considered, but they turned out to be equivalent to Vopěnka cardinals.

Woodin cardinals and determinacy

$ZF+AD$, $ZFC+AD^{L(\mathbb{R})}$ and $ZFC+$"the non-stationary ideal over $\omega_1$ is $\omega_1$-dense" and $ZFC+$"there exists infinitely many Woodin cardinals" are equiconsistent.

Under $ZF+AD$, the model $HOD^{L(\mathbb{R})}\models ZFC+\Theta^{L(\mathbb{R})}$ is a Woodin cardinal. [3] gives many generalization of this result.

If there exists infinitely many Woodin cardinals with a measurable above them all, then $AD^{L(\mathbb{R})}$. If there assumtion that there is a measurable above those Woodins is removed, one still has projective determinacy.

In fact projective determinacy is equivalent to "for every $n<\omega$, there is a fine-structural, countably iterable inner model $M$ such that $M\models ZFC+$"there exists $n$ Woodin cardinals".

For every $n$, if there exists $n$ Woodin cardinals with a measurable above them all, then all $\mathbf{\Sigma}^1_{n+1}$ sets are determined.

$\mathbf{\Pi}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is a countable ordinal $\delta$ such that $\delta$ is a Woodin cardinal in some inner model of $ZFC$ containing $x$.

$\mathbf{\Delta}^1_2$-determinacy is equivalent to "for every $x\in\mathbb{R}$, there is an inner model M such that $x\in M$ and $M\models ZFC+$"there is a Woodin cardinal".

$ZFC+$ lightface $\Delta^1_2$-determinacy implies that there many $x$ such that $HOD^{L[x]}\models ZFC+\omega_2^{L[x]}$ is a Woodin cardinal.

$Z_2+$ lightface $\Delta^1_2$-determinacy is conjectured to be equiconsistent with $ZFC+$"Ord is Woodin", where "Ord is Woodin" is expressed as an axiom scheme and $Z_2$ is second-order arithmetic.

$Z_3+$ lightface $\Delta^1_2$-determinacy is provably equiconsistent with $NBG+$"Ord is Woodin" where $NBG$ is Von Neumann–Bernays–Gödel set theory and $Z_3$ is third-order arithmetic.